Interference Between an Arduino Chip And A ESP-01 WiFi Module

My thoughts are heading in the direction of supply rail decoupling, ground planes, etc. I don’t have an ESP8266, but as far as I’m aware, one can work happily with an emonTx, in the form of the emonESP

It would certainly be worth looking into decoupling all the supply rails, because although you’ve said it’s electromagnetic coupling, at least some of it might actually be conducted. What you’ve done with the ribbon cable is add a bit of inductance and capacitance, which might just have been enough to allow the ESP to work. So it might be worth adding some filtering to the supply to the ESP. I think that, in the long term, if you replace the prototype breadboard with a custom pcb that observes all the standard layout precautions, then you’re much more likely to have success.